Combinatorics in the group of parity alternating permutations

We call a permutation parity alternating, if its entries assume even and odd integers alternately. Parity alternating permutations form a subgroup of the symmetric group. This paper deals with such permutations classified by two permutation statistics; the numbers of ascents and inversions. It turns out that they have a close relationship to signed Eulerian numbers. The approach is based on a study of the set of permutations that are not parity alternating. It is proved that the number of even permutations is equal to that of odd ones among such permutations with the same ascent number. Hence signed Eulerian numbers can be described by parity alternating permutations. Divisibility properties for the cardinalities of certain related sets are also deduced.